Interactivate

Events and Set Operations

Mentor: We can look at a dice game to understand an
event in probability. If a player wins when a six-sided die rolls a 1 or 2, we can say: "The
event of the player winning happens in two outcomes out of six." Sometimes people draw pictures for
events, circling or highlighting all the outcomes for each event.

Event A: the player wins.

Event B: the player loses.

Mentor: Let us consider one more game as an example. In this game, four players share a twelve-sided
die. Player A (Anton) wins if the die shows 1, 4, 6, or 12. Player B (Boris) wins if the die
shows 2, 3, 4, 7 or 12. Player C (Chris) wins if the die shows 1, 4, 8, 9, or 12. Player D
(Dorothy) wins if none of the other players win. I will describe several events to demonstrate
how convenient the diagrams can be. First try to find the probabilities without looking at the
diagrams: this way you will see which of them are hard to find without the diagrams. Find the
probabilities of the events listed in the table. The corresponding diagrams and the answers
are also in the table. Instead of writing: "The probability of Event A is .33" we can write
simply P(A)=.33 The answers follow from counting the outcomes, out of twelve total.

Event A: Player A wins

P(A)=4/12=1/3

Event B: Player B wins

P(B)=5/12

Event C: Player C wins

P(C)=5/12

Event D: Player D wins

P(D)=3/12=1/4

Event E: Player A or Player B wins (there is a special short notation for this:
E = A U B which reads: "Event E is equal to the union of Events A and B)

P(E)=7/12

Event F: Player B wins but Player C does not win (the special notation for this is
F = B\C which reads: "Event F is equal to Event B minus Event C")

P(F)=3/12=1/4

Event G: Both Player A and Player C win (the notation for that is
which reads: "Event G is equal to the intersection of Events A and C")

P(G)=3/12=1/4

Event H: Player D does not win (the notation here is
H = D
C which reads: "Event H is equal to the complement of Event D").